Finite Group Databasebeta

← #20273710

Subgroups — #20273710

Order 6784.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811220C211yesmaximal, characteristic
2#16810953C211yesmaximal, characteristic, complemented
3#16810936C211yesmaximal, characteristic
4#16810359C211yesmaximal, characteristic, complemented
5#16810356C211yesmaximal, characteristic
6#16810349C211yesmaximal, characteristic, complemented
7C53×#398C211yesmaximal, characteristic, complemented
8#13133212—21no—
9#13133207—21no—
10#13133202—21no—
11#13133200—21no—
12#13133196C2211yescharacteristic
13#13133195—21no—
14#13133191C2211yescharacteristic
15C4×C22⋊D53—21no—
16#13133150—21no—
17#13133148—21no—
18#13133140—21no—
19#13133137—21no—
20C22⋊#88322C2211yescharacteristic
21C22⋊#88322C2211yescharacteristic
22C2×C4⋊C4×C53—21no—
23C2×C4⋊C4×C53C2211yescharacteristic
24C2×C4⋊C4×C53C2211yescharacteristic
25#95×C53—21no—
26#95×C53C2211yescharacteristic
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53D411yescharacteristic
30C22×C4×C53D411yescharacteristic
31C22×C4×C53C2311yescharacteristic
32C2×C22⋊D53—21no—
33C2×D212D411yescharacteristic
34C2×C4×D53—21no—
35C2×Dic106D411yescharacteristic
36C4⋊C4×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53—42no—
41C22⋊Dic53—21no—
42#11719921—21no—
43#11719921—21no—
44#11719921—21no—
45#11719921—21no—
46C4⋊Dic53D411yescharacteristic
47C4⋊Dic53D411yescharacteristic
48#11719919—21no—
49#11719919—21no—
50#11719919—21no—
51#11719919—21no—
52C4×Dic53—21no—
53C2×#88322—21no—
54C2×#88322—21no—
55C23×C53C2×D411yescharacteristic
56C22×D53—21no—
57C2×C4×C53—22no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53—41no—
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C22⋊D53—41no—
72C22⋊D53—41no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76D212—41no—
77C4×D53—81no—
78Dic106—41no—
79#88322—41no—
80#88322—41no—
81C22×C53—21no—
82C22×C53—21no—
83C22×C53#12011yescharacteristic
84C22×C53#13511yescharacteristic
85C22×C53AGL(1,ℤ/8)11yescharacteristic
86C2×D53—41no—
87C2×D53—41no—
88C2×D53—41no—
89C4×C53—21no—
90C4×C53—21no—
91C4×C53—42no—
92C4×C53—41no—
93C4×C53—41no—
94C4×C53—81no—
95Dic53—41no—
96Dic53—41no—
97Dic53—41no—
98#1609—531nomaximal, Sylow
99C2×C53—21no—
100C2×C53—21no—
101C2×C53#44911yescharacteristic
102C2×C53#45611yescharacteristic
103C2×C53#45811yescharacteristic
104D53—81no—
105#539—531no—
106#483—531no—
107#480—531no—
108#398D5311yescharacteristic, complemented
109#328—531no—
110#327—531no—
111#324—531no—
112C53#160911yescharacteristic, complemented, Sylow
113#126—1061no—
114#125—1061no—
115#123—1061no—
116#123—1061no—
117C22⋊Q8—531no—
118C22⋊Q8—1061no—
119#121—531no—
120C4×D4—1061no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2×D5311yescharacteristic
123C2×C4⋊C4C2×D5311yescharacteristic
124#107—1061no—
125#106—1061no—
126#103—1061no—
127#102—1061no—
128C22⋊C8—531no—
129C22⋊C8—531no—
130#95—21no—
131#95C2×D5311yescharacteristic
132C2×Q8—531no—
133C2×D4—531no—
134C2×D4—1061no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—1061no—
138C22×C4C22⋊D5311yescharacteristic
139C22×C4C22⋊D5311yescharacteristic
140C22×C4C22×D5311yescharacteristic
141C2×C8—1061no—
142C2×C8—1061no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—42no—
148C4⋊C4—531no—
149C4⋊C4—531no—
150C4⋊C4—1061no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
153C4⋊C4—1061no—
154C22⋊C4—1061no—
155C22⋊C4—1061no—
156C22⋊C4—1061no—
157C22⋊C4—1061no—
158C22⋊C4—1061no—
159C42—1061no—
160C23—1061no—
161C23C2×C22⋊D5311yescharacteristic
162Q8—2121no—
163D4—2121no—
164D4—2121no—
165D4—2121no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—1061no—
179C2×C4—1061no—
180C2×C4—1061no—
181C2×C4—4241no—
182C2×C4D4×D5311yescharacteristic
183C2×C4D4×D5311yescharacteristic
184C8—2121no—
185C8—2121no—
186C22—21no—
187C22—21no—
188C22—2121no—
189C22—2121no—
190C22—2121no—
191C22#1313318511yescharacteristic
192C22#1313323711yescharacteristic
193C22#1313324311yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—81no—
200C4—2121no—
201C4—2121no—
202C4—2121no—
203C2—21no—
204C2—21no—
205C2—4241no—
206C2#1681076611yescharacteristic
207C2#1681080511yescharacteristic
208C2#1681081211yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#398C2
socleC22×C53#120
hypercenter#398D53
hypocenterC53#1609
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #16810953C2semidirect
4. #16810359C2semidirect
6. #16810349C2semidirect
7. C53×#398C2semidirect
108. #398D53semidirect
112. C53#1609semidirect